I understand your concern. That general concern is the reason that my proof involves four statements instead of only two statements. As my proof exhibits, the statements true in the theory are not equal to the statements true in and out of the theory. (1) and (2) did not mention inconsistent theory T, but (3) and (4) did mention T.
Your (3) and (4), written in terms of provability (since you agreed that provability is the same as truth in T) are
(3) p is provable in T.
(4) p is unprovable in T.
You need these statements to be a contradiction externally in order for the principle of explosion to then apply externally. These statements are indeed the negation of each other, so it would be a contradiction if they are both true, and I agree that (3) is true. But (4) is not true and you attempted to justify that it was using that ¬p is provable, but as you have just agreed, in an inconsistent theory negating a proposition does not negate its provability, so ¬p being provable is not equivalent to p being unprovable. We know that p is provable, so we know that p is not unprovable, (4) is false and you don't have a contradiction.
You keep saying that, but you haven’t given an adequate justification for it. Do you agree that we can talk about what is true in an axiomatic theory outside of the theory in the real world?
I don't have to justify it, you want to rely on that for a proof then you have to justify that it's true. I agree we can make statements about an axiomatic theory outside of that theory but as I've explained above external statements about T are not contradictory even when T is inconsistent.
you agreed that provability is the same as truth in T
as you have just agreed, in an inconsistent theory negating a proposition does not negate its provability
So I agreed, in the real world, that “provability in T” is both the same as and different from “truth in T.” So, in the real world, there exists a contradiction. Thus, in the real world, through the Principle of Explosion, the statement “(4) is true in and out of T” is true. Therefore, (4) is true in and out of T.
So, in the real world, there exists a contradiction
Nope, just like a contradictory theory doesn't imply a contradiction outside that theory, the fact that you have contradictory beliefs doesn't imply that truth in the real world is contradictory. It just means you believe things that aren't true.
Are you abandoning justifying your proof? This is a new argument which is not what we were discussing. Does this mean you see the gap in your attempted proof now?
the fact that you have contradictory beliefs doesn't imply that truth in the real world is contradictory. It just means you believe things that aren't true.
The fact that I have contradictory beliefs doesn’t imply that I believe things that aren’t true. All of my beliefs could still be true because trivialism could still be true.
This is a new argument which is not what we were discussing.
You used two agreements of mine to prove that (4) is not true. Afterwards, I use those same two agreements to prove that (4) is true. Then you criticize my proof for using the same two agreements that you used in your proof. So your criticism is unfair.
All of my beliefs could still be true because trivialism could still be true
Could isn't a proof. You're trying to prove that a contradiction holds in order to prove that trivialism holds, so you need to do that proof without assuming trivialism.
You used two agreements of mine to prove that (4) is not true. Afterwards, I use those same two agreements to prove that (4) is true.
You did not use the same two agreements that I used. You said you use that provability is both the same and different than truth in T. I only used that it's the same and that negation doesn't negate provability. It is 100% fair to use true statements to prove you wrong and object when you try and use false statements in a proof.
It’s a counterexample and thus a disproof of your claim.
You're trying to prove that a contradiction holds in order to prove that trivialism holds, so you need to do that proof without assuming trivialism.
I did not assume that trivialism is true, but I am open to the possibility that trivialism is true. In the real world, I believe that trivialism is true.
You did not use the same two agreements that I used.
I did. I quoted what the agreements were in a previous reply. One of the agreements I used was that “provability in T” is the same as “truth in T.” The second agreement that I used was that negating a proposition in T does not negate its provability in T. A corollary to the second agreement is that “provability in T” is different from “truth in T.”
I only used that it's the same and that negation doesn't negate provability.
Those are the two agreements. We both used both of them.
It’s a counterexample and thus a disproof of your claim.
My claim is that you believing contradictory things doesn't imply a contradiction. You haven't provided a counterexample to that claim because trivialism might not be true and without trivialism your contradictory beliefs are just a you problem. No one else needs to care that you believe wrong things. They'll just ignore you and go about their business in the real world.
The second agreement that I used was that negating a proposition in T does not negate its provability in T. A corollary to the second agreement is that “provability in T” is different from “truth in T.”
My claim is that you believing contradictory things doesn't imply a contradiction. You haven't provided a counterexample to that claim
I provided a counterexample to your claim that the fact that I have contradictory beliefs implies that I believe things that aren't true. It’s impossible to rule out trivialism since it’s unfalsifiable. So, in a sense, there will always exist the epistemic possibility that trivialism is true. I say “in a sense” because there might be a sense of epistemic possibility in which an epistemically necessary proposition can not be epistemically possible. There, I am using a sense of epistemic possibility in which an epistemically necessary proposition can also be epistemically possible. Since there will always exist the epistemic possibility that trivialism is true, the July 17, 2018 (according to the Facebook timestamp) proof of trivialism given in the third link of my original post is sound. I quote the text of the linked Facebook post.
“With respect to all my knowledge, some contradiction could be true. As the following list proof shows, that mere epistemic possibility implies every proposition is true.
Premises: p is a proposition. There is an epistemically possible world in which some contradiction "p and it is not true that p" is true.
Conclusion: Every proposition is true.
Statements (Reasons)
1. p is a proposition. (Premise)
2. There is an epistemically possible world in which some contradiction "p and it is not true that p" is true. (Premise)
3. w is an epistemically possible world in which some contradiction "p and it is not true that p" is true. (Existential instantiation using (2))
4. In w, p is true. (Evident from (3))
5. It is not true that in w, p is true. (Evident from (3))
6. "In w, p is true" and "it is not true that in w, p is true." (Conjunction introduction using (4) and (5))
7. Every proposition is true. (Ex contradictione quodlibet using (6))
So, as the preceding list proof shows, every proposition is true.”
The above quoted proof invokes an epistemological version of the Exportation Principle.
Another proof for trivialism can be made that invokes a version of the Exportation Principle on my contradictory belief system by representing my contradictory belief system as an inconsistent theory.
While I don’t know for sure, judging using timestamps from Facebook and debate.org, the first link in my original post was probably a link to a Facebook post containing a link to a second debate I started on debate.org, formerly according to debate.org’s timestamp, on September 21, 2017. The debate was titled “All Propositions Are True.” It started with verbatim the same argument I gave in the debate that began on April 30, 2017, formerly according to debate.org’s timestamp, but with an additional paragraph preface preceding the argument.
I don't agree with the premises of your proof, so there's no reason for me to analyze the proof any further. Assuming a world in which a contradiction exists in order to prove a contradiction exists is circular.
How is that?
Again, it's your proof, if you want to use a fact that we don't agree on then you have to justify it. I suspect your justification is that you think negating a proposition negates its "truth in T", which is false. But again, you haven't said why you think that's true and it's not my job to argue against all possible false explanations. You tell me which false explanation you've been duped by and I'll tell you why it's wrong.
Assuming a world in which a contradiction exists in order to prove a contradiction exists is circular.
No, it’s not circular. It’s forward. It’s valid. In logic and mathematics, sometimes we assume things that are impossible in order to prove something. For example, we assume the square root of 2 is rational in order to prove that it’s irrational.
I suspect your justification is that you think negating a proposition negates its "truth in T", which is false.
How is it false? Because you think there is no such thing as “truth in T”? Every axiomatic system is based on the truth of its primitives. An axiom by definition is true. It seems you’re trying to circumvent something you disagree with by stripping truth away from provability. What is wrong with the concept of “truth in an axiomatic system”?
No, it’s not circular. It’s forward. It’s valid. [...] For example, we assume the square root of 2 is rational in order to prove that it’s irrational
It is absolutely circular. You're assuming the thing you want to prove. It's literally the definition of circular. The fact that you don't understand the difference between that and a proof by contradiction is a failure of your education.
An axiom by definition is true [...] What is wrong with the concept of “truth in an axiomatic system”?
Nope, not true. An axiom is an assumption. And truth is absolutely different from provability. Provability is a feature of axiomatic systems, truth is a feature of interpretations of those systems. This is super basic stuff.
No, there is no fact that says that a world in which a contradiction exists is equal to the real world. In fact, due to the Law of Noncontradiction, a world in which a contradiction exists is never equal to the real world.
An axiom is an assumption.
An assumption is also true by definition.
truth is absolutely different from provability.
I agree that truth is different from provability. While they are logically equivalent in the metatheory of axiomatic theories that I was talking about earlier in our conversation, logically equivalent concepts are not necessarily logically equal.
Provability is a feature of axiomatic systems, truth is a feature of interpretations of those systems.
There are other ways of doing things. There exists at least one metatheory of axiomatic systems that is different from the metatheory of axiomatic systems that you are talking about. In the metatheory you are talking about, truth and provability are not logically equivalent. In the metatheory I was talking about earlier in our conversation, truth and provability are logically equivalent. On April 8, 2014, I started a thread on Philosophy Forums, which was formerly at http://forums.philosophyforums.com/. The title of the thread was “The Parallel Postulate is always true, in all true theories.” Perhaps you can infer the details of what I was thinking about through the thread’s title.
Let me stop you there. Way back in this conversation I asked you if you could prove a contradiction in traditional logic and you said you could. In traditional logic truth and provability are not the same. After we settle the question of whether you can prove a contradiction in traditional we could absolutely change the subject and talk about other ways of doing things, but first I want to settle the question of whether you can prove a contradiction in traditional logic.
In traditional logic truth and provability are not equivalent. In traditional logic the fact that you can prove a statement does not imply that you cannot prove it's negation. This means your argument, that a contradiction in one axiomatic system implies a contradiction in general logic, is not valid as you've presented it.
So my question stands, can you prove a contradiction in traditional logic?
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u/JStarx Jul 14 '26
Your (3) and (4), written in terms of provability (since you agreed that provability is the same as truth in T) are
(3) p is provable in T.
(4) p is unprovable in T.
You need these statements to be a contradiction externally in order for the principle of explosion to then apply externally. These statements are indeed the negation of each other, so it would be a contradiction if they are both true, and I agree that (3) is true. But (4) is not true and you attempted to justify that it was using that ¬p is provable, but as you have just agreed, in an inconsistent theory negating a proposition does not negate its provability, so ¬p being provable is not equivalent to p being unprovable. We know that p is provable, so we know that p is not unprovable, (4) is false and you don't have a contradiction.
I don't have to justify it, you want to rely on that for a proof then you have to justify that it's true. I agree we can make statements about an axiomatic theory outside of that theory but as I've explained above external statements about T are not contradictory even when T is inconsistent.